In the following, we give an example application of Theorem  with respect to the pumping lemma for regular languages (see e.g. [1]}). Aside from defining an interesting superclass of the regular languages itself, there are many reasons to be interested in notions of pumping. For example, when considering (in)expressibility questions (even beyond the regular languages), as well as part of a strategy for producing satisfiability results in the context of length constraints or other restrictions. We use the pumping lemma for regular languages because it is well known, but the ideas are easily adapted to other useful notions of pumping and closure properties more generally. We recall first this lemma.
[[1]}]
Let \(L\)  be a regular language. Then there exists a constant \(c\)  such that for every \(w \in L\)  with \(|w| > c\) , there exist \(x,y,z\)  such that
(i) \(|xy| < c\) , and
(ii) \(w = xyz\) , and
(iii) \(xy^nz \in L\)  for all \(n \in \mathbb {N}_0\) .
